+6 If n and k are positive integers such that \(5<\frac {n}{k}<6\), then what is the smallest possible value of \(\frac{\mathop{\text{lcm}}[n,k]}{\gcd(n,k)}\) ?
We know that gcd(n,k) divides both n and k, so gcd(n,k)n and gcd(n,k)k are integers. Also, lcm(n,k) is a multiple of both n and k, so gcd(n,k)lcm(n,k) is an integer. Therefore, [\frac{\text{lcm}(n,k)}{\gcd(n,k)} = \frac{n}{\gcd(n,k)} \cdot \frac{k}{\gcd(n,k)}.]We are given that 5
Since 5 < n/k < 6, we know that k is a divisor of n but k < n < 6k. Therefore, the smallest possible value of k is 2 and the smallest possible value of n is 11. In this case, lcm(n,k) = 22 and gcd(n,k) = 2, so lcm(n,k)/gcd(n,k) = 11. Therefore, the smallest possible value of lcm(n,k)/gcd(n,k) is 11.
